American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Quotients of orders in cyclic algebras and space-time codes
  • AMC Home
  • This Issue
  • Next Article
    The cross-correlation distribution of a $p$-ary $m$-sequence of period $p^{2k}-1$ and its decimated sequence by $\frac{(p^{k}+1)^{2}}{2(p^{e}+1)}$
November  2013, 7(4): 425-440. doi: 10.3934/amc.2013.7.425

On the dual of (non)-weakly regular bent functions and self-dual bent functions

Ayça Çeşmelioǧlu 1, , Wilfried Meidl 2, and  Alexander Pott 1,

1. 

Faculty of Mathematics, Otto-von-Guericke University, Universitätsplatz 2, 39106, Magdeburg, Germany, Germany
2. 

MDBF, Sabanci University, Orhanlı, Tuzla 34956, İstanbul, Turkey

Received  July 2012 Revised  March 2013 Published  October 2013

For weakly regular bent functions in odd characteristic the dual function is also bent. We analyse a recently introduced construction of non-weakly regular bent functions and show conditions under which their dual is bent as well. This leads to the definition of the class of dual-bent functions containing the class of weakly regular bent functions as a proper subclass. We analyse self-duality for bent functions in odd characteristic, and characterize quadratic self-dual bent functions. We construct non-weakly regular bent functions with and without a bent dual, and bent functions with a dual bent function of a different algebraic degree.
Keywords: Fourier transform., self-dual bent functions, Duals of bent functions.
Mathematics Subject Classification: Primary: 11T71, 94A60, 06E30; Secondary: 11T2.
Citation: Ayça Çeşmelioǧlu, Wilfried Meidl, Alexander Pott. On the dual of (non)-weakly regular bent functions and self-dual bent functions. Advances in Mathematics of Communications, 2013, 7 (4) : 425-440. doi: 10.3934/amc.2013.7.425
References:
[1]

C. Carlet, On the secondary constructions of resilient and bent functions,, in Coding, (2004), 3.   Google Scholar

[2]

C. Carlet, P. Charpin and V. Zinoviev, Codes, bent functions and permutations suitable for DES-like cryptosystems,, Des. Codes Cryptogr., 15 (1998), 125.  doi: 10.1023/A:1008344232130.  Google Scholar

[3]

C. Carlet, L. E. Danielsen, M. G. Parker and P. Solé, Self-dual bent functions,, Int. J. Inform. Coding Theory, 1 (2010), 384.  doi: 10.1504/IJICOT.2010.032864.  Google Scholar

[4]

C. Carlet, H. Dobbertin and G. Leander, Normal extensions of bent functions,, IEEE Trans. Inform. Theory, 50 (2004), 2880.  doi: 10.1109/TIT.2004.836681.  Google Scholar

[5]

A. Çeşmelioǧlu, G. McGuire and W. Meidl, A construction of weakly and non-weakly regular bent functions,, J. Comb. Theory Ser. A, 119 (2012), 420.  doi: 10.1016/j.jcta.2011.10.002.  Google Scholar

[6]

A. Çeşmelioǧlu and W. Meidl, A construction of bent functions from plateaued functions,, Des. Codes Cryptogr., 66 (2013), 231.  doi: 10.1007/s10623-012-9686-2.  Google Scholar

[7]

Y. M. Chee, Y. Tan and X. D. Zhang, Strongly regular graphs constructed from $p$-ary bent functions,, J. Algebr. Comb., 34 (2011), 251.  doi: 10.1007/s10801-010-0270-4.  Google Scholar

[8]

Y. Edel and A. Pott, On the equivalence of nonlinear functions,, in NATO Advanced Research Workshop on Enhancing Cryptographic Primitives with Techniques from Error Correcting Codes, (2009), 87.   Google Scholar

[9]

K. Garaschuk and P. Lisoněk, On ternary Kloosterman sums modulo 12,, Finite Fields Appl., 14 (2008), 1083.  doi: 10.1016/j.ffa.2008.07.002.  Google Scholar

[10]

F. Göloǧlu, G. McGuire and R. Moloney, Ternary Kloosterman sums modulo $18$ using Stickelberger's theorem,, in Proceedings of SETA 2010 (eds. C. Carlet and A. Pott), (2010), 196.  doi: 10.1007/978-3-642-15874-2_16.  Google Scholar

[11]

T. Helleseth and A. Kholosha, Monomial and quadratic bent functions over the finite fields of odd characteristic,, IEEE Trans. Inform. Theory, 52 (2006), 2018.  doi: 10.1109/TIT.2006.872854.  Google Scholar

[12]

T. Helleseth and A. Kholosha, New binomial bent functions over the finite fields of odd characteristic,, IEEE Trans. Inform. Theory, 56 (2010), 4646.  doi: 10.1109/TIT.2010.2055130.  Google Scholar

[13]

T. Helleseth and A. Kholosha, Crosscorrelation of m-sequences, exponential sums, bent functions and Jacobsthal sums,, Cryptogr. Commun., 3 (2011), 281.  doi: 10.1007/s12095-011-0048-0.  Google Scholar

[14]

X. D. Hou, Classification of self dual quadratic bent functions,, Des. Codes Cryptogr., 63 (2012), 183.  doi: 10.1007/s10623-011-9544-7.  Google Scholar

[15]

K. P. Kononen, M. J. Rinta-aho and K. O. Väänänen, On integer values of Kloosterman sums,, IEEE Trans. Inform. Theory, 56 (2010), 4011.  doi: 10.1109/TIT.2010.2050806.  Google Scholar

[16]

N. G. Leander, Monomial bent functions,, IEEE Trans. Inform. Theory, 52 (2006), 738.  doi: 10.1109/TIT.2005.862121.  Google Scholar

[17]

R. Lidl and H. Niederreiter, Finite Fields, Second edition,, Cambridge Univ. Press, (1997).   Google Scholar

[18]

Y. Tan, A. Pott and T. Feng, Strongly regular graphs associated with ternary bent functions,, J. Comb. Theory Ser. A, 117 (2010), 668.  doi: 10.1016/j.jcta.2009.05.003.  Google Scholar

[19]

Y. Tan, J. Yang and X. Zhang, A recursive approach to construct $p$-ary bent functions which are not weakly regular,, in Proceedings of IEEE International Conference on Information Theory and Information Security, (2010), 156.   Google Scholar

show all references

References:
[1]

C. Carlet, On the secondary constructions of resilient and bent functions,, in Coding, (2004), 3.   Google Scholar

[2]

C. Carlet, P. Charpin and V. Zinoviev, Codes, bent functions and permutations suitable for DES-like cryptosystems,, Des. Codes Cryptogr., 15 (1998), 125.  doi: 10.1023/A:1008344232130.  Google Scholar

[3]

C. Carlet, L. E. Danielsen, M. G. Parker and P. Solé, Self-dual bent functions,, Int. J. Inform. Coding Theory, 1 (2010), 384.  doi: 10.1504/IJICOT.2010.032864.  Google Scholar

[4]

C. Carlet, H. Dobbertin and G. Leander, Normal extensions of bent functions,, IEEE Trans. Inform. Theory, 50 (2004), 2880.  doi: 10.1109/TIT.2004.836681.  Google Scholar

[5]

A. Çeşmelioǧlu, G. McGuire and W. Meidl, A construction of weakly and non-weakly regular bent functions,, J. Comb. Theory Ser. A, 119 (2012), 420.  doi: 10.1016/j.jcta.2011.10.002.  Google Scholar

[6]

A. Çeşmelioǧlu and W. Meidl, A construction of bent functions from plateaued functions,, Des. Codes Cryptogr., 66 (2013), 231.  doi: 10.1007/s10623-012-9686-2.  Google Scholar

[7]

Y. M. Chee, Y. Tan and X. D. Zhang, Strongly regular graphs constructed from $p$-ary bent functions,, J. Algebr. Comb., 34 (2011), 251.  doi: 10.1007/s10801-010-0270-4.  Google Scholar

[8]

Y. Edel and A. Pott, On the equivalence of nonlinear functions,, in NATO Advanced Research Workshop on Enhancing Cryptographic Primitives with Techniques from Error Correcting Codes, (2009), 87.   Google Scholar

[9]

K. Garaschuk and P. Lisoněk, On ternary Kloosterman sums modulo 12,, Finite Fields Appl., 14 (2008), 1083.  doi: 10.1016/j.ffa.2008.07.002.  Google Scholar

[10]

F. Göloǧlu, G. McGuire and R. Moloney, Ternary Kloosterman sums modulo $18$ using Stickelberger's theorem,, in Proceedings of SETA 2010 (eds. C. Carlet and A. Pott), (2010), 196.  doi: 10.1007/978-3-642-15874-2_16.  Google Scholar

[11]

T. Helleseth and A. Kholosha, Monomial and quadratic bent functions over the finite fields of odd characteristic,, IEEE Trans. Inform. Theory, 52 (2006), 2018.  doi: 10.1109/TIT.2006.872854.  Google Scholar

[12]

T. Helleseth and A. Kholosha, New binomial bent functions over the finite fields of odd characteristic,, IEEE Trans. Inform. Theory, 56 (2010), 4646.  doi: 10.1109/TIT.2010.2055130.  Google Scholar

[13]

T. Helleseth and A. Kholosha, Crosscorrelation of m-sequences, exponential sums, bent functions and Jacobsthal sums,, Cryptogr. Commun., 3 (2011), 281.  doi: 10.1007/s12095-011-0048-0.  Google Scholar

[14]

X. D. Hou, Classification of self dual quadratic bent functions,, Des. Codes Cryptogr., 63 (2012), 183.  doi: 10.1007/s10623-011-9544-7.  Google Scholar

[15]

K. P. Kononen, M. J. Rinta-aho and K. O. Väänänen, On integer values of Kloosterman sums,, IEEE Trans. Inform. Theory, 56 (2010), 4011.  doi: 10.1109/TIT.2010.2050806.  Google Scholar

[16]

N. G. Leander, Monomial bent functions,, IEEE Trans. Inform. Theory, 52 (2006), 738.  doi: 10.1109/TIT.2005.862121.  Google Scholar

[17]

R. Lidl and H. Niederreiter, Finite Fields, Second edition,, Cambridge Univ. Press, (1997).   Google Scholar

[18]

Y. Tan, A. Pott and T. Feng, Strongly regular graphs associated with ternary bent functions,, J. Comb. Theory Ser. A, 117 (2010), 668.  doi: 10.1016/j.jcta.2009.05.003.  Google Scholar

[19]

Y. Tan, J. Yang and X. Zhang, A recursive approach to construct $p$-ary bent functions which are not weakly regular,, in Proceedings of IEEE International Conference on Information Theory and Information Security, (2010), 156.   Google Scholar

[1]

Sihem Mesnager, Fengrong Zhang, Yong Zhou. On construction of bent functions involving symmetric functions and their duals. Advances in Mathematics of Communications, 2017, 11 (2) : 347-352. doi: 10.3934/amc.2017027
[2]

Jacques Wolfmann. Special bent and near-bent functions. Advances in Mathematics of Communications, 2014, 8 (1) : 21-33. doi: 10.3934/amc.2014.8.21
[3]

Claude Carlet, Fengrong Zhang, Yupu Hu. Secondary constructions of bent functions and their enforcement. Advances in Mathematics of Communications, 2012, 6 (3) : 305-314. doi: 10.3934/amc.2012.6.305
[4]

Sihem Mesnager, Fengrong Zhang. On constructions of bent, semi-bent and five valued spectrum functions from old bent functions. Advances in Mathematics of Communications, 2017, 11 (2) : 339-345. doi: 10.3934/amc.2017026
[5]

Ayça Çeşmelioğlu, Wilfried Meidl. Bent and vectorial bent functions, partial difference sets, and strongly regular graphs. Advances in Mathematics of Communications, 2018, 12 (4) : 691-705. doi: 10.3934/amc.2018041
[6]

Bram van Asch, Frans Martens. Lee weight enumerators of self-dual codes and theta functions. Advances in Mathematics of Communications, 2008, 2 (4) : 393-402. doi: 10.3934/amc.2008.2.393
[7]

Samir Hodžić, Enes Pasalic. Generalized bent functions -sufficient conditions and related constructions. Advances in Mathematics of Communications, 2017, 11 (3) : 549-566. doi: 10.3934/amc.2017043
[8]

Claude Carlet, Juan Carlos Ku-Cauich, Horacio Tapia-Recillas. Bent functions on a Galois ring and systematic authentication codes. Advances in Mathematics of Communications, 2012, 6 (2) : 249-258. doi: 10.3934/amc.2012.6.249
[9]

Joan-Josep Climent, Francisco J. García, Verónica Requena. On the construction of bent functions of $n+2$ variables from bent functions of $n$ variables. Advances in Mathematics of Communications, 2008, 2 (4) : 421-431. doi: 10.3934/amc.2008.2.421
[10]

Jyrki Lahtonen, Gary McGuire, Harold N. Ward. Gold and Kasami-Welch functions, quadratic forms, and bent functions. Advances in Mathematics of Communications, 2007, 1 (2) : 243-250. doi: 10.3934/amc.2007.1.243
[11]

Kanat Abdukhalikov, Sihem Mesnager. Explicit constructions of bent functions from pseudo-planar functions. Advances in Mathematics of Communications, 2017, 11 (2) : 293-299. doi: 10.3934/amc.2017021
[12]

Xiwang Cao, Hao Chen, Sihem Mesnager. Further results on semi-bent functions in polynomial form. Advances in Mathematics of Communications, 2016, 10 (4) : 725-741. doi: 10.3934/amc.2016037
[13]

Yanfeng Qi, Chunming Tang, Zhengchun Zhou, Cuiling Fan. Several infinite families of p-ary weakly regular bent functions. Advances in Mathematics of Communications, 2018, 12 (2) : 303-315. doi: 10.3934/amc.2018019
[14]

Natalia Tokareva. On the number of bent functions from iterative constructions: lower bounds and hypotheses. Advances in Mathematics of Communications, 2011, 5 (4) : 609-621. doi: 10.3934/amc.2011.5.609
[15]

Sihong Su. A new construction of rotation symmetric bent functions with maximal algebraic degree. Advances in Mathematics of Communications, 2019, 13 (2) : 253-265. doi: 10.3934/amc.2019017
[16]

Wenying Zhang, Zhaohui Xing, Keqin Feng. A construction of bent functions with optimal algebraic degree and large symmetric group. Advances in Mathematics of Communications, 2020, 14 (1) : 23-33. doi: 10.3934/amc.2020003
[17]

Chunming Tang, Maozhi Xu, Yanfeng Qi, Mingshuo Zhou. A new class of $ p $-ary regular bent functions. Advances in Mathematics of Communications, 2019, 0 (0) : 0-0. doi: 10.3934/amc.2020042
[18]

Gabriele Nebe, Wolfgang Willems. On self-dual MRD codes. Advances in Mathematics of Communications, 2016, 10 (3) : 633-642. doi: 10.3934/amc.2016031
[19]

Masaaki Harada, Akihiro Munemasa. Classification of self-dual codes of length 36. Advances in Mathematics of Communications, 2012, 6 (2) : 229-235. doi: 10.3934/amc.2012.6.229
[20]

Stefka Bouyuklieva, Anton Malevich, Wolfgang Willems. On the performance of binary extremal self-dual codes. Advances in Mathematics of Communications, 2011, 5 (2) : 267-274. doi: 10.3934/amc.2011.5.267

2018 Impact Factor: 0.879

Tools

Metrics

  • PDF downloads (13)
  • HTML views (0)
  • Cited by (10)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences